Computation of Bessel and Airy Functions and of Related Gaussian Quadrature Formulae
Procedures are described for the high-precision calculation of the modified Bessel function Kν(x), 0 < ν < 1, and the Airy function Ai(x), for positive arguments x, as pre-requisites for generating Gaussian quadrature rules having these functions as weight function.
Unable to display preview. Download preview PDF.
- 1.M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NBS Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, DC, 1964.Google Scholar
- 2.W. Gautschi, How and how not to check Gaussian quadrature formulae, BIT, 23 (1983), pp. 209–216.Google Scholar
- 3.W. Gautschi, Algorithm 726: ORTHPOL-A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software, 20 (1994), pp. 21–62.Google Scholar
- 4.W. Gautschi, Orthogonal polynomials: applications and computation, Acta Numerica, 5 (1996), pp. 45–119.Google Scholar
- 7.I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, San Diego, CA, 2000.Google Scholar
- 8.D.W. Lozier and F. W. J. Olver, Numerical evaluation of special functions, in Mathematics of Computation 1943–1993: A half-century of computational mathematics, Vancouver, BC, 1993, W. Gautschi, ed., Proc. Sympos. Appl. Math., Vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 79-125.Google Scholar
- 9.Z. Schulten, D. G. M. Anderson, and R. G. Gordon, An algorithm for the evaluation of the complex Airy functions, J. Comput. Phys., 31 (1979), pp. 60–75.Google Scholar
- 10.N. M. Temme, On the numerical evaluation of the modified Bessel function of the third kind, J. Comput. Phys., 19 (1975), pp. 324–337.Google Scholar
- 11.G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1958.Google Scholar
- 12.R. Wong, Quadrature formulas for oscillatory integral transforms, Numer. Math., 39 (1982), pp. 351–360.Google Scholar